Subgroups — C₃₁×#17964 #21474723

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3171	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1594	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1592	C₂	1	2	yes	maximal, complemented
6	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1569	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1496	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1495	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#529	C₂²	1	1	yes	characteristic
15	C₃₁×#501	C₄	1	1	yes	characteristic
16	C₃₁×#496	C₄	1	1	yes	characteristic
17	C₃₁×#490	C₄	1	1	yes	characteristic
18	C₃₁×#489	C₄	1	1	yes	characteristic
19	C₃₁×#469	C₄	1	2	yes	—
20	C₃₁×#468	C₄	1	2	yes	—
21	C₃₁×#446	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₄×D₈×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#436	—	2	1	no	—
26	C₃₁×#436	C₂²	1	2	yes	—
27	C₃₁×#436	C₂²	1	1	yes	characteristic
28	C₃₁×#435	—	2	1	no	—
29	C₈×D₄×C₃₁	—	2	1	no	—
30	C₈×D₄×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#433	—	2	1	no	—
34	C₃₁×#432	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#408	—	2	1	no	—
38	C₃₁×#405	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	—
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
45	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
47	C₃₁×#322	C₂²	1	2	yes	—
48	C₈²×C₃₁	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
52	C₃₁×#131	—	4	4	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
63	C₃₁×#126	—	2	1	no	—
64	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#124	—	2	1	no	—
66	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
67	C₃₁×#123	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
69	C₃₁×#121	—	2	1	no	—
70	C₃₁×#121	C₂×C₄	1	2	yes	—
71	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×D₄×C₃₁	C₂³	1	2	yes	—
77	C₃₁×#117	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	C₂×C₄	1	2	yes	—
80	C₃₁×#106	C₂×C₄	1	2	yes	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	—	2	1	no	—
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#103	C₂×C₄	1	2	yes	—
89	C₃₁×#102	C₂×C₄	1	2	yes	—
90	C₃₁×#102	C₂×C₄	1	2	yes	—
91	C₃₁×#102	C₂×C₄	1	2	yes	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	1	no	—
97	C₃₁×C₂²⋊C₈	—	2	1	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₃₁×#97	—	2	1	no	—
100	C₃₁×#97	C₂×C₄	1	1	yes	characteristic
101	C₃₁×#97	C₂×C₄	1	1	yes	characteristic
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
104	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	C₄×C₈×C₃₁	C₂³	1	2	yes	—
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	#41×C₃₁	—	4	2	no	—
110	#41×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
124	SD₁₆×C₃₁	—	2	4	no	—
125	D₈×C₃₁	—	2	4	no	—
126	M₁₆×C₃₁	—	2	4	no	—
127	M₁₆×C₃₁	—	4	2	no	—
128	M₁₆×C₃₁	—	4	4	no	—
129	M₁₆×C₃₁	—	4	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	4	1	no	—
136	C₂×C₈×C₃₁	—	4	2	no	—
137	C₂×C₈×C₃₁	—	4	4	no	—
138	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₃₁	—	2	1	no	—
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
154	C₂²⋊C₄×C₃₁	—	2	1	no	—
155	C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂²⋊C₄×C₃₁	—	2	1	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₃₁	#41	1	1	yes	characteristic
163	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
164	#17964	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₃₁	—	2	1	no	—
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
168	Q₈×C₃₁	—	2	2	no	—
169	Q₈×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	2	2	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	2	1	no	—
197	C₈×C₃₁	—	2	1	no	—
198	C₈×C₃₁	—	2	1	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₈×C₃₁	—	4	1	no	—
201	C₈×C₃₁	—	4	2	no	—
202	C₈×C₃₁	—	4	1	no	—
203	C₈×C₃₁	#131	1	4	yes	complemented
204	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
205	#3171	C₂×C₃₁	1	1	yes	characteristic
206	#2966	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1594	C₂×C₃₁	1	2	yes	complemented
209	#1592	C₂×C₃₁	1	2	yes	complemented
210	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
211	C₈×D₈	C₂×C₃₁	1	2	yes	complemented
212	#1569	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1496	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1495	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
216	C₂²×C₃₁	—	2	1	no	—
217	C₂²×C₃₁	—	2	1	no	—
218	C₂²×C₃₁	—	2	1	no	—
219	C₂²×C₃₁	—	4	1	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	1	no	—
225	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₃₁	—	2	1	no	—
227	C₄×C₃₁	—	2	1	no	—
228	C₄×C₃₁	—	2	1	no	—
229	C₄×C₃₁	—	2	1	no	—
230	C₄×C₃₁	—	4	1	no	—
231	C₄×C₃₁	—	4	1	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	C₂×#131	1	2	yes	—
235	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₄×C₃₁	#575	1	1	yes	characteristic
237	C₂×#131	—	2	1	no	—
238	#532	C₂²×C₃₁	1	1	yes	characteristic
239	#529	C₂²×C₃₁	1	1	yes	characteristic
240	#501	C₄×C₃₁	1	1	yes	characteristic
241	#496	C₄×C₃₁	1	1	yes	characteristic
242	#490	C₄×C₃₁	1	1	yes	characteristic
243	#489	C₄×C₃₁	1	1	yes	characteristic
244	#469	C₄×C₃₁	1	2	yes	—
245	#468	C₄×C₃₁	1	2	yes	—
246	#446	C₂²×C₃₁	1	1	yes	characteristic
247	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
249	C₄×D₈	C₂²×C₃₁	1	2	yes	—
250	#436	—	2	1	no	—
251	#436	C₂²×C₃₁	1	2	yes	—
252	#436	C₂²×C₃₁	1	1	yes	characteristic
253	#435	—	2	1	no	—
254	C₈×D₄	—	2	1	no	—
255	C₈×D₄	—	2	1	no	—
256	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
257	C₈×D₄	C₂²×C₃₁	1	2	yes	—
258	#433	—	2	1	no	—
259	#432	C₂²×C₃₁	1	1	yes	characteristic
260	#431	C₂²×C₃₁	1	1	yes	characteristic
261	#423	C₂²×C₃₁	1	1	yes	characteristic
262	#408	—	2	1	no	—
263	#405	—	2	1	no	—
264	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
266	#335	C₂²×C₃₁	1	2	yes	—
267	#334	C₂²×C₃₁	1	2	yes	—
268	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
269	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
270	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
271	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
272	#322	C₂²×C₃₁	1	2	yes	—
273	C₈²	C₂²×C₃₁	1	2	yes	—
274	C₂×C₃₁	—	4	1	no	—
275	C₂×C₃₁	—	8	2	no	—
276	C₂×C₃₁	—	8	1	no	—
277	C₂×C₃₁	#2931	1	1	yes	characteristic
278	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₂×C₃₁	#2955	1	1	yes	characteristic
280	C₂×#41	—	2	1	no	—
281	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
282	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
283	#131	—	4	4	no	—
284	C₂×M₁₆	—	2	1	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	1	no	—
287	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
291	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
292	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
293	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
294	#126	—	2	1	no	—
295	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
296	#124	—	2	1	no	—
297	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
298	#123	—	2	1	no	—
299	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
300	#121	—	2	1	no	—
301	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
302	#121	C₂×C₄×C₃₁	1	2	yes	—
303	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₄×D₄	C₂³×C₃₁	1	2	yes	—
308	#117	—	2	1	no	—
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	#107	C₂×C₄×C₃₁	1	2	yes	—
311	#106	C₂×C₄×C₃₁	1	2	yes	—
312	C₄⋊C₈	—	2	1	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
319	#103	C₂×C₄×C₃₁	1	2	yes	—
320	#102	C₂×C₄×C₃₁	1	2	yes	—
321	#102	C₂×C₄×C₃₁	1	2	yes	—
322	#102	C₂×C₄×C₃₁	1	2	yes	—
323	C₂²⋊C₈	—	2	1	no	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	—	2	2	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	#97	—	2	1	no	—
330	#97	—	2	1	no	—
331	#97	C₂×C₄×C₃₁	1	1	yes	characteristic
332	#97	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₄×C₈	—	2	1	no	—
334	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
335	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
336	C₄×C₈	C₂³×C₃₁	1	2	yes	—
337	C₄×C₈	C₂³×C₃₁	1	2	yes	—
338	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₃₁	#17964	1	1	yes	characteristic, direct product factor, Sylow
341	#41	—	4	2	no	—
342	#41	—	4	2	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
356	SD₁₆	—	2	4	no	—
357	D₈	—	2	4	no	—
358	M₁₆	—	2	4	no	—
359	M₁₆	—	4	2	no	—
360	M₁₆	—	4	4	no	—
361	M₁₆	—	4	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	1	no	—
368	C₂×C₈	—	4	2	no	—
369	C₂×C₈	—	4	4	no	—
370	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
371	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
372	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
382	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
383	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	2	1	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	—	2	1	no	—
391	C₄²	—	2	1	no	—
392	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
393	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₄²	#41×C₃₁	1	1	yes	characteristic
395	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂³	—	2	1	no	—
397	C₂³	—	2	2	no	—
398	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
399	Q₈	—	2	2	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	2	2	no	—
403	D₄	—	2	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	4	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
426	C₈	—	2	2	no	—
427	C₈	—	2	1	no	—
428	C₈	—	2	1	no	—
429	C₈	—	2	1	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	1	no	—
432	C₈	—	4	2	no	—
433	C₈	—	4	1	no	—
434	C₈	C₃₁×#131	1	4	yes	complemented
435	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	1	no	—
445	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	C₂×C₃₁×#131	1	2	yes	—
455	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
456	C₄	C₃₁×#575	1	1	yes	characteristic
457	C₂	—	4	1	no	—
458	C₂	—	8	2	no	—
459	C₂	—	8	1	no	—
460	C₂	C₃₁×#2931	1	1	yes	characteristic
461	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
462	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3171	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2966	C₂	semidirect
3. C₃₁×#2931	C₂	semidirect
4. C₃₁×#1594	C₂	semidirect
5. C₃₁×#1592	C₂	semidirect
6. C₈×SD₁₆×C₃₁	C₂	semidirect
7. C₈×D₈×C₃₁	C₂	semidirect
8. C₃₁×#1569	C₂	semidirect
9. C₃₁×#1496	C₂	semidirect
10. C₃₁×#1495	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
164. #17964	C₃₁	direct (N × H)
203. C₈×C₃₁	#131	semidirect
206. #2966	C₂×C₃₁	semidirect
207. #2931	C₂×C₃₁	semidirect
208. #1594	C₂×C₃₁	semidirect
209. #1592	C₂×C₃₁	semidirect
210. C₈×SD₁₆	C₂×C₃₁	semidirect
211. C₈×D₈	C₂×C₃₁	semidirect
212. #1569	C₂×C₃₁	semidirect
213. #1496	C₂×C₃₁	semidirect
214. #1495	C₂×C₃₁	semidirect
215. C₈×M₁₆	C₂×C₃₁	semidirect
340. C₃₁	#17964	direct (N × H)
434. C₈	C₃₁×#131	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3171	2
C₃₁×#2966	2
C₃₁×#2931	2
C₃₁×#1594	2
C₃₁×#1592	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
C₃₁×#1569	2
C₃₁×#1496	2
C₃₁×#1495	2
C₈×M₁₆×C₃₁	2
#17964	31

Subgroups — C31×#17964 #21474723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17964 #21474723